Suppose we have a function that is analytic on the open unit disk.
Suppose we have a continuous function on the boundary of the disk that maps each point on the boundary of the disk to its conjugate.
Suppose that near the boundary of the unit disk, the values of our analytic function approach the values of our continuous function.
Can I say that the analytic function has to be identically the conjugation function? I'm looking find a contradiction like this. (In this case, I know that the conjugation function is not analytic.)
Complex analysis isn't my strongest subject. I'll appreciate any help you can give me!